A dynamic approach to the tendency of industries to cluster

From (6) we see that:

(9)

( ⁿι

Z = ∑

V i=¹

1-¹ ъ ■ - A σ-¹

^x≈1i^σ +∑ x2Γ

i =¹ )

Making a substitution of (7) and (8) into (9) leaves us with:

(10)

Z = ^xrn^σ ∑

¹¹¹

j V Pj 1 )

Collecting the terms in (10) that relate to input x_i₁₁ and simplifying leads to:

(11)

Z = ^xi11 Pn ∑ P j1^σV j

⁺^xi11 Pil Σ ⁽Pj2 ^t)
V ^j

¹^-^σ

)

Aσ-1

This can be written as:
(12) Z = x_n1 Ph P ^{- -}

( ⁿ1 ⁿ2 A 1-n

^, wh^ere^p = I ∑ p}1 ^σ⁺∑⁽p_j21⁾¹^-^σ I

Due to the symmetric fashion in which the

V i =1 i =1 )

producer service varieties enter the sub-production function of the manufacturing firm,
this expression of P can be simplified to read:
⁽¹³⁾^p = (n 1 p 1^-^σ⁺ n2⁽ p21⁾¹^-^σ )¹^-^σ

, which the same as Equation (2). To see that P is the cost function, we solve for input
demand:
(14) x_fl1 = ZPi-^σ P p

Letting the cost function be evaluated at a production of Z=1 leads to:

√^-^σ

(¹⁵) c = ∑ P11^-^σP^σ +∑ (Pi21)¹^-^σ P^σ = P^{σ (}n 1 p1^-^σ + n2(P21)¹^-^σ⁾¹-σ = P^σP¹-^σ = P

i i

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